formulario Calculo
June 20, 2019 | Author: theband | Category: N/A
Short Description
formulario basico y util para calculo diferencial e integral...
Description
Formul ormular ario io de Prec Prec´ ´ alcu alculo lo.. 1.
5. Leyes de los logaritmos. a ) loga (P Q) = loga (P ) P ) + loga (Q) P b ) loga = loga (P ) P ) loga (Q) Q
Los Numeros. u ´meros.
1. Leyes de los exponentes y radicales. a ) am an = am+n d )
a b
n
n
=
g ) a1/n = j ) j )
b ) (am )n = amn
a bn
e)
√a
√ab = √a √b n
n
n
a = am−n n a
h ) am/n =
n
k )
n
a = b
c ) loga (Qn ) = n loga (Q)
c ) (ab) ab)n = an bn
m
f ) f ) a−n =
√am √a √b
e ) loga (ax ) = x
√
n
l )
n
√
m
n
d ) aloga (x) = x
1 an
i ) am/n = ( n a)
n
a=
f ) f ) loga (1) = 0
m
g ) aloga (a) = 1
√a
mn
h ) log(x log(x) = log10 (x)
2. Productos Productos Notables. Notables.
i ) ln(x ln(x) = loge (x)
− y) = x2 − y2 b ) Binomio al Cuadrado: (x (x ± y )2 = x2 ± 2xy + y 2 c ) Binomio al Cubo: (x (x ± y )3 = x3 ± 3x2 y + 3xy 3xy 2 ± y 3
a ) Binomios Binomios Conjugados: Conjugados: (x (x + y )(x )(x
j ) j ) Cambi Cambioo de base base::
logb (Q) logb (a)
log loga (Q) =
2. Soluc Solucio ione ness Exact Exactas as de ecu ecuac acioiones Algebraicas
2
d ) d ) (x + y ) = x2 + 2 xy + y 2 e ) (x
−
− y)2 = x2 − 2 xy + y2 3
f ) f ) (x + y ) = x3 + 3 x2 y + 3 xy 2 + y 3 g ) (x
6. Soluciones Exactas de Ecuaciones Algebraicas.
− y)3 = x3 − 3 x2y + 3 xy2 − y3
a ) La Ecuaci´ on on Cuadr Cu adr´ ´ atica ati ca:: ax2 + bx + c = 0 tiene soluciones: b b2 4ac x= 2a 2 El n´ umero umero b 4ac se llama discriminante de la ecuaci´ on. on. i) Si b2 4ac > 0 las ra´ ra´ıces son reales y diferentes. 2 ii) Si b 4ac = 0 las ra´ ra´ıces son reales e iguales. 2 iii) Si b 4ac < 0 las la s ra´ıces ıces son so n complejas compl ejas conjugaconjug adas. b ) Para la Ecuaci´ on o n C´ ubica: ubica: x3 + ax2 + bx + c = 0 sean:
4
h ) (x + y ) = x4 + 4 x3 y + 6 x2 y 2 + 4 xy3 + y 4 i ) i ) (x
− ±√ −
− y)4 = x4 − 4 x3y + 6 x2y2 − 4 xy3 + y4 5
j ) j ) (x + y ) = x5 + 5 x4y + 10 x3y 2 + 10 x2y 3 + 5 xy4 + y 5
−
− y)5 = x5 − 5 x4y + 10 x3y2 − 10 x2y3 + 5 xy4 − y5 3. Teorema del Binomio. Sea n ∈ N, entonces: k ) k ) (x
n
n
(x + y ) =
r=0
Nota:
− − −
n n−r r x y r
n n! = n Cr = r r!(n !(n r)!
−
Q=
3b
9
4. Factores Notables. a ) Diferencia Diferencia de Cuadrados: Cuadrados: x2
− y2 = (x ( x + y )(x )(x − y ) b ) Suma de Cubos: x3 + y 3 = (x + y)(x )(x2 − xy + y 2 ) c ) Diferencia Diferencia de Cubos: Cubos: x3 − y 3 = (x ( x − y )(x )(x2 + xy + y 2 ) d ) d ) Trinomio Cuadrado Perfecto: x2 ± 2xy+ xy + y 2 = (x± y)2 e ) x2 − y 2 = (x ( x − y ) (x + y ) 3− 3 f ) f ) x y = (x ( x − y ) x2 + xy + y 2 g ) x3 + y 3 = (x ( x + y ) x2 − xy + y 2 h ) x4 − y 4 = (x ( x − y ) (x + y ) x2 + y 2 i ) i ) x5 − y 5 = (x ( x − y ) x4 + x3 y + x2 y 2 + xy3 + y 4 j ) j ) x5 + y 5 = (x ( x + y ) x4 − x3 y + x2 y 2 − xy3 + y 4 k ) k ) x6 −y 6 = (x ( x − y) (x + y ) x2 + xy + y 2 x2 − xy + y 2 l ) l ) x4 + x2 y 2 + y4 = x2 + xy + y 2 x2 − xy + y 2 m ) x4 + 4 y 4 = x2 − 2 xy + 2 y 2 x2 + 2 xy + 2 y 2
1
S =
− a2 ,
R=
3
R+
9ab
Q3 + R2 ,
54
T =
Entonces las soluciones son: a x1 =S + S + T 3 S + S + T a x2 = + + 2 3
−
x3 =
− −
S + S + T a + 2 3
− 27 27cc − 2a3
−
− 3
R
√ (S − T ) T ) 3 2 √ (S − T ) T ) 3 2
Q3 + R2
i i
El n´ umero umero Q3 +R2 se llama discriminante de la ecuaci´ on. on. i) Si Q3 + R2 > 0, hay una ra´ ra´ız real y dos son s on complejas conjugadas. ii) Si Q3 + R2 = 0, las ra´ ra´ıces son reales y por lo menos dos son iguales. iii) Si Q3 + R2 < 0, las l as ra r a´ıces son reales y diferentes. di ferentes.
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Formul ormular ario io de C´ alcu alculo lo.. Derivadas. En este formulario: k, c R son constantes reales, f = f ( f (x), u = u(x) y v = v(x) son funciones que dependen de x.
∈
Funciones Funcion es Trigono rig onom´ m´ etricas etr icas:: Funci´ on:
Su Derivada:
f = sen(u sen(u)
f ′ = cos(u cos(u) u′
f = cos(u cos(u)
f ′ =
f = tan(u tan(u) f = csc(u csc(u)
F´ ormulas ormul as B´ asicas asi cas:: Funci´ on:
Su Derivada:
f = k
f ′ = 0
f = sec(u sec(u) f = cot(u cot(u)
·
− sen(u sen(u) · u′ f ′ = sec2 (u) · u′ f ′ = − csc(u csc(u) cot( cot(u u) · u′ f ′ = sec(u sec(u) tan( tan(u u) · u′ f ′ = − csc2 (u) · u′
Linealidad de la derivada: f = k u
· f = u ± v f = k · u ± c · v
f ′ = k u′
Funciones uncion es Trigonom´ rigon om´ etricas etricas Inversas:
·
f ′ = u′
± v′ f ′ = k · u′ ± c · v′
Regla del Producto: f ′ = u v′ + v u′
f = u v
·
·
·
Funci´ on: f = arc sen( sen(u)
Su Derivada: u′ ′ f = ; 1 u2
f = arc cos( cos(u)
f ′ =
− √1u− u2 ; |u| < 1
f = arctan(u arctan(u)
f ′ =
u′ 1 + u2
f = arccsc(u arccsc(u)
f ′ = −
f = arcsec(u arcsec(u)
f ′ =
u √ ; |u| > 1 u u2 − 1
f = arccot(u arccot(u)
f ′ =
− 1 +u u2 ; |u| > 1
√ −
Regla del Cociente: Cociente: f =
u v
f ′ =
v · u′ − u · v′ v2
Regla de la Cadena (Composici´ on on de funciones) f = u(x) v(x)
◦
f ′ = [u(v(x))]′ v′ (x)
·
Regla de la Potencia: f = vn f = k vn
·
′
′
u √ u u2 − 1
′
′
f ′ = n vn−1 v′
· · f ′ = k · n · vn−1 · v′
Funciones Exponenciales: u
|u| < 1
f = e
f ′ = eu · u′
f = au
f ′ = au ln(a ln(a) u′
·
Funciones Funcio nes Logar Log ar´ ´ıtmica ıtm icas: s:
′
·
Funciones Hiperb´ olicas: olicas: Funci´ on:
Su Derivada:
f = senh(u senh(u)
f ′ = cosh(u cosh(u) u′
f = cosh(u cosh(u) f = tanh(u tanh(u) f = csch(u csch(u)
· f ′ = senh(u senh(u) · u′ f ′ = sech2 (u) · u′ f ′ = −csch(u csch(u) coth( coth(u u) · u′
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Definici´ on on 1. Ecuac Ecuaci´ i´ on en Variables Separadas. on Consideremos la ecuaci´on on con forma est´andar: andar: M ( M (x)dx )dx + N ( N (y )dy )dy = 0
(1)
La soluci´ on se obtiene integrando directamente: on
M ( M (x)dx )dx +
N ( N (y )dy )dy = C
Definici´ on on 2. Ecuac Ecuaci´ i´ on en Variables Separables. on Las siguientes dos ecuaciones, son ecuaciones en variables separables.
)dx + M 2 (x)N 2 (y )dy )dy = 0 M 1 (x)N 1 (y)dx
dy = f ( f (x)g (y ) dx
(2)
(3)
Para determinar la soluci´on on de la Ec.(2), se divide la ecuaci´on on entre: M 2 (x)N 1 (y), para reducirla a la ecuaci´ on en variables separadas: on
La soluci´ on de la Ec.(3), se obtiene al dividir entre on g (y) y multiplicar por dx d x, para reducirla a la ecuaci´on on en variables separadas:
M 1 (x) N 2 (y ) dx + dy = 0 M 2 (x) N 1 (y )
1 dy = f ( f (x)dx )dx g (y )
ahora s´olo olo se integra directamente:
Definici´
M 1 (x) dx + M 2 (x)
3. Ec
N 2 (y ) dy = C N 1 (y )
ci´ cion ´ on Lineal.
ahora s´olo olo se integra directamente:
1 dy = g (y )
f ( f (x)dx )dx + C
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Definici´ on on 5. Ecuaciones Ecuaciones Exactas Exactas o en Diferenci Diferenciales ales Totales. Totales. Consideramos la ecuaci´on: on: M (x, y )dx )dx + N ( N (x, y )dy )dy = 0
(8)
donde se cumple: M y = N x . La soluci´on on se obtiene de calcular: i) ii)
u=
M (x, y )dx )dx,
calculamos: uy
iii)
v = [N ( N (x, y )
− uy ]dy ]dy
iv) iv)
La solu soluci ci´ on o ´n general impl´ıcita ıcita es: u + v = C
Definici´ on on 6. Factor Integran Integrante. te. Consideremos la ecuaci´on: on: M (x, y )dx )dx + N ( N (x, y )dy )dy = 0
(9)
donde M y = N x . Para determinar la soluci´on on de esta ecuaci´on, on, se tiene que reducir a una ecuaci´on on exacta; exa cta; as´ı que q ue primero se debe calcular uno de los dos posibles factores integrantes:
M y N x dx N
1) µ(x) = e
−
N x My dy M
2) µ(y ) = e
−
segundo se multiplica la Ec.(9) por el factor integrante que exista y se obtiene la ecuaci´on exacta: exacta:
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B) Con el cambio de variable v = xy . La Ec.(11) Ec.(11) se reduce a la ecuaci´on on en variables separadas: dy M ( M (v, 1) + dv = 0 y N ( N (v, 1) + vM ( vM (v, 1)
la cual cual se inte integra gra dire direct ctam amen ente te
la soluci´ on de la Ec.(11) se obtiene al sustituir nuevamente v por on
x y
dy + y
M ( M (v, 1) dv = C N ( N (v, 1) + vM ( vM (v, 1)
en el resultado de la integral.
La Ec.(12) Ec.(12) se reduce a la ecuaci´on on en variables separadas: dv 1 f (v,1) v, 1)
−v
=
dy y
la cual se integra directamente
1 f ( f (v,1) v,1)
la soluci´ on de la Ec.(12) se obtiene al sustituir nuevamente v por on
x y
y2
···
yn
−v
=
dy + C y
en el resultado de la integral.
I. Wronskiano. y1
dv
Rengl´ on on de las funciones.
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Caso iii. Ra´ Ra´ıces Conjugadas Conjug adas Complejas, Complej as, y3 (x). Sean m1 = α1 β 1 i, m2 = α2 β 2 i, m3 = α3 β 3 i , . . . las ra´ ra´ıces complejas conjugadas de (15), entonces, otra parte de yh (x) se escribe como:
±
±
±
y3 (x) = eα1 x C 1 cos(β cos(β 1 x) + C 2 sen(β sen(β 1 x) +
eα2 x C 3 cos(β cos(β 2 x) + C 4 sen(β sen(β 2 x) +
eα3 x C 5 cos(β cos(β 3 x) + C 6 sen(β sen(β 3 x) +
Nota: Nota: Obs´ ervese ervese que se toma el valor positivo positivo de β en todos las casos.
···
(18)
Caso iv. Ra´ Ra´ıces Conjuga C onjugadas das Complejas Co mplejas Repetidas, Repeti das, y4 (x). Sean m1 = α βi = m2 = α βi = m3 = α βi = las ra´ ra´ıces conjugadas complejas repetidas rep etidas de (15), entonces, otra parte de yh (x) se escribe como:
±
±
±
···
y4 (x) = eαx C 1 cos(βx cos(βx)) + C 2 sen(βx sen(βx)) +
eαx C 3 cos(βx cos(βx)) + C 4 sen(βx sen(βx)) +
x
2 x
eαx C 5 cos(βx cos(βx)) + C 6 sen(βx sen(βx)) +
···
(19)
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Segundo. Caso i. Ecuaci´ on on de segundo orden. La soluci´ solucion o´n particular tiene la forma: y p(x) = u1 y1 + u2 y2 donde: u′1 = u′2 =
−g(x)y2 ,
u1 =
g (x)y1 , W [ W [y1 , y2 ]
u2 =
W [ W [y1 , y2 ]
g (x)y2 dx W [ W [y1 , y2]
−
g (x)y1 dx W [ W [y1 , y2]
Caso ii. Ecuaci´ on on de tercer orden. La soluci´ solucion o´n particular tiene la forma: y p (x) = u1 y1 + u2 y2 + u3 y3 donde: ′
′
g (x)[y )[y2 y3 − y3 y2 ] u′1 = , W [ W [y y y ]
u1 =
′
′
g (x)[y )[y2 y3 y3 y2 ] dx W [ W [y y y ]
−
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• Primer Teorema de Traslaci´on on o de Desplazamiento: at f (t)} = F ( F (s − a) L {e f ( Primero identificamos el valor de a y se calcula L {f ( f (t)} = F ( F (s). Segundo se calcula F ( F (s) s=s−a , y as´ as´ı se cumple que at L {e f ( f (t)} = F ( F (s − a). • Funci´on on Escal´on on Unitario Unitario de Heaviside, Heaviside, denotada denotada como U (t − a) o H (t − a). 0, 0 ≤ t ≤ a; H (t − a) = U (t − a) = 1, t ≥ a. • Funci´on on por partes en t´ erminos erminos la l a funci´on on escal´on on unitario. Sea f 1 (t) 0 ≤ t ≤ a f 2 (t) a ≤ t < b f ( f (t) = f 3 (t) b ≤ t < c f 4 (t) t ≥ c
entonces:
f ( f (t) = f 1 (t)U (t) + f 2 (t)
− f 1(t) U (t − a) +
f 3 (t)
− f 2(t) U (t − b) +
f 4 (t)
− f 3(t) U (t − c)
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